THE CONVEX INTERSECTION BODY OF A CONVEX BODY

Glasgow Math. J.53(2011) 523–534.Glasgow Mathematical Journal Trust 2011.

doi:10.1017/S0017089511000103.

THE CONVEX INTERSECTION BODY OF A CONVEX BODY

MATHIEU MEYER

Universit´e Paris-Est - Marne-la-Vall´ee, Laboratoire d’Analyse et de Math´ematiques Appliqu´ees (UMR 8050), Cit´e Descartes, 5 Bd Descartes, Champs-sur-Marne, 77454 Marne-la-Vall´ee cedex 2, France

e-mail: mathieu.meyer@univ-mlv.fr

and SHLOMO REISNER†

Department of Mathematics, University of Haifa, Haifa 31905, Israel e-mail: reisner@math.haifa.ac.il

(Received 2 April 2010; revised 29 September 2010; accepted 30 November 2010;

first published online 10 March 2011)

Abstract. LetLbe a convex body in⺢ⁿandzan interior point ofL. We associate withLandza new, convex and centrally symmetric, bodyCI(L,z). This generalizes the classicalintersection body I(L,z) (whose radial function atu∈Sⁿ⁻¹is the volume of the hyperplane section ofLthroughz, orthogonal tou).CI(L,z) coincides with I(L,z) if and only if L is centrally symmetric aboutz. We study the properties of CI(L,z).

2010Mathematics Subject Classification.52A20.

1. Introduction. LetLbe a convex body in⺢ⁿcontaining 0 in its interior. The intersection bodyI(L) of L, defined by its radial functionρI(L) on the sphereSⁿ⁻¹, which is

ρI(L)(u)=vol(L∩u^⊥) and the cross-section bodyC(L) ofL, defined by

ρC(L)(u)=max

t vol((L∩(tu+u^⊥))

are not, in general, convex bodies, although they are identical and, moreover, convex in the case whenLis centrally symmetric. This follows from Brunn–Minkowski theorem (see [21, p. 309]) and from Busemann’s theorem (see [3]). We introduce here a new convex body associated withL, generalizing both the intersection body and the cross- section body. More precisely, we define theconvex intersection body CI(L) ofLby its radial function

ρCI(L)(u)= min

z∈Pu(L^∗g(L))vol

P_u(L^∗g(L))_∗z

, (1)

†Both authors were supported in part by the France-Israel Research Network Program in Mathematics contract #3-4301.

whereg(L) denotes here the centroid ofL, and ifEis a linear subspace of⺢ⁿandM is a convex body inE, we define forz∈E,

M^∗z= {y∈E;y−z,x−z ≤1 for everyx∈M}.

Thus (1) means: first apply duality with respect to the pointg(L), then project ontou^⊥, finally apply duality with respect tozand minimize the (n−1)-dimensional volume overz.

In Section 2, we shall attach to any convex bodyKin⺢ⁿa bodyJ(K) constructed with the help of its projections, and prove that it is always convex. In Section 3, we prove thatCI(L) is convex, and we study the inclusions

CI(L)⊂I(L)⊂C(L),

wheng_L=0. Finally, in Section 4, a few open problems are listed, with proposed ideas concerning some of them.

NOTATIONS. Forx,y∈⺢ⁿ, we denote byx,ythe canonical scalar product in⺢ⁿ,

|x|denotes the Euclidean norm defined by it. Ifuandvare two non-zero vectors in⺢ⁿ that are not orthogonal to one another, we define_w,u^⊥ :⺢ⁿ→u^⊥to be the projection parallel towontou^⊥= {x∈⺢ⁿ;x,u =0}, we denote

P_u=u,u^⊥.

IfLis a subset of⺢ⁿ, let [L] be the affine subspace of⺢ⁿthat it spans. IfBis a convex subset of⺢ⁿ, we denote its k-dimensional volume by vol(B) (wherek=dim[B]). By conv (A), we denote the closed convex hull ofA.

ForLa convex set in⺢ⁿandz∈[L], we denote the polar body ofLwith respect tozby

L^∗z= {y∈[L];y−z,x−z ≤1 for everyx∈L}.

It is well known that the functionz→vol(L^∗z) is strictly convex on the relative interior ofLand tends to+∞aszapproaches the realtive boundary ofLin [L] (see [17,20]). It follows that it reaches its minimum at a unique points(L)∈int(L). This point is called theSantal´o point of L. We shall denote

L^∗s:=L^∗s(L).

Moreover,s(L) is also characterized as the unique pointz∈int(L), which is the centroid ofL^∗z(see [20] and also [21, p. 419]). Let us denote byg(M) the centroid of a convex bodyMin [M], and set

M^∗g=M^∗g(M). One has

L^∗s=Mif and only ifM^∗g=L.

Observe that, in general, s(L)=g(L) (see the recent [18], where a lower bound to how far apart these two points can be is given). Finally if 0∈int(M), we shall write M^∗ =M^∗0, so thatM^∗x=x+(M−x)^∗for everyx∈int(M).

We adopt the following notation: ifKis a star body with respect to 0, we denote

||x||K =inf{λ >0;x∈λK}

to be thegaugeofK. Then foru∈Sⁿ⁻¹, ρK(u)= 1

||u||K

is theradial functionofK.

2. A convexity theorem.

THEOREM1. If K is a convex body in ⺢ⁿ, let N_K: ⺢ⁿ→⺢+ be defined by the formula

NK(u)= 1

vol((P_uK)^∗s) = 1

min_z∈u⊥vol((P_uK)^∗z)for u∈Sⁿ⁻¹,

and extended to all⺢ⁿby NK(ru)=rNK(u)for r≥0and u∈Sⁿ⁻¹. Then NKis a norm on⺢ⁿ.

Before proving Theorem 1, we need some preliminary results.

DEFINITION. Let v∈Sⁿ⁻¹, B be a bounded subset of ⺢ⁿ and V:B→⺢ be a bounded map. Theshadow system(L_t),t∈[a,b] of convex bodies in⺢ⁿ, withdirection the vectorv, withbasisthe setBand withspeed the functionV, is the family of convex bodies

L_t=conv{b+tV(b)v;b∈B}, fort∈[a,b].

For the sake of completeness, we prove the following result, which appears in [23] and is used, for example, in [4,5].

PROPOSITION 2.Let K be a convex body in ⺢ⁿ. Then, for u, v∈Sⁿ⁻¹, such that u, v =0, the family Lt=_u+tv,u^⊥K, t∈⺢, is a shadow system of convex bodies in u^⊥, in the directionv.

Proof.To simplify notation, one may suppose that u=en andv=en−1, where e1, . . . ,enis an orthonormal basis of⺢ⁿ (we shall write⺢^jfor [e1, . . . ,ej]). Then, for allt∈⺢,

u+tv,u^⊥K = {X+ze_n−1;X∈⺢ⁿ⁻²,X+ze_n−1+r(e_n+te_n−1)∈Kfor somer∈⺢}

= {X+zen−1;X∈⺢ⁿ⁻²,X+(z+rt)en−1+ren∈Kfor somer∈⺢}

= {X+(x−rt)e_n−1; X∈⺢ⁿ⁻²,r∈⺢such thatX+xe_n−1+re_n∈K}

= {U−rten−1; (U,r)∈PuK×⺢such thatU+ren∈K}.

ForU∈PuK, define

I(U)= {r∈⺢;U+ren∈K}.

ThenI(U)=[a(U),b(U)] is a closed interval of⺢. Define alsox(U)∈⺢such that U−x(U)e_n−1,e_n−1 =0,

and let

D₁= {U∈P_uK;x(U)∈⺡}andD₂= {U∈P_uK;x(U)∈⺢\⺡}.

DefineV :PuK→⺢by

v(U)= −b(U) ifU∈D₁andv(U)= −a(U) ifU∈D₂.

By the continuity of the two concave functions−a,b:Pu(K)→⺢, it is easy to see that for everyt∈⺢

u+tv,u^⊥K=conv{U+tV(U)e_n;U∈P_uK}.

REMARK. The converse assertion of Proposition 2 is true : every shadow system L_t in ⺢ⁿ can be represented as L_t=u+tv,u^⊥(K) with an appropriate convex body K⊂⺢ⁿ⁺¹andu, v∈Sⁿ. This was shown, for example, in [4].

The following result was proved in [16].

THEOREM3.Let t∈[a,b]→Lt be a shadow system in⺢ⁿ, then the functionφ:

⺢ⁿ→⺢, defined by the formula φ(t)= 1

vol((Lt)^∗s) = 1

minzvol((Lt)^∗z), is convex.

LEMMA4.Let N:⺢ⁿ→⺢₊satisfy

r

_{N(x)>0for x=0,}

r

_{N(αx)= |α|N(x)for everyα∈⺢and x∈⺢}ⁿ_,

r

_{for all u, v∈S}ⁿ⁻¹_{such thatu, v =0, t→N(u+tv)}is convex on⺢. Then N is a norm on⺢ⁿ.

Proof. Let us show that N(x+y)≤N(x)+N(y) for every x,y∈⺢ⁿ\ {0}. Let α= |_|x|^x +_|y|^y|andβ= |_|x|^x −_|y|^y|,u= ¹_α(_|x|^x +_|y|^y ) andv= _β¹(_|x|^x −_|y|^y). We may suppose thatα=0 andβ=0. Thenu, v∈Sⁿ⁻¹,u, v =0 and

x= |x|

2 (αu+βv), y= |y|

2 (αu−βv). We get

N(x+y)= α(|x| + |y|)

2 N

u+β(|x| − |y|) α(|x| + |y|)v

= α(|x| + |y|)

2 N(u+(λt+(1−λ)s)v),

where λ= _|x|+|^|x|_y|, t= ^β_α and s= −^β_α. Under the assumption of the lemma, we get

N(x+y)≤ α(|x| + |y|)

2 (λN(u+tv)+(1−λ)N(u+sv))

= α 2

|x|N

u+β αv

+ |y|N

u−β

αv

=N(x)+N(y).

Proof of Theorem 1.In view of Lemma 4, we need to prove that t→g_u,v(t)= N(u+tv) is convex, wheneveru, v∈S_n−1 satisfyu, v =0. It is easy to see that for anyt∈⺢,P_u+tvKis an affine image ofu+tv,u^⊥Kand satisfies

vol(Pu+tvK)= 1

√1+t²vol(u+tv,u^⊥K).

Hence

min

z∈{u+tv}^⊥vol((P_u+tvK)^∗z)=

1+t²min

z∈u^⊥vol((u+tv,u^⊥K)^∗z)) It follows that

N(u+tv)= |u+tv|

min_z∈{u+tv}⊥vol((P_u+tvK)^∗z)) = 1

min_z∈u⊥vol((u+tv,u^⊥K)^∗z) . Now by Proposition 2,t→u+tv,u^⊥Kis a shadow system on⺢and thus by Theorem 3, g_u,vis convex on⺢.

REMARKS. (1) IfKis centrally symmetric (and centred at 0), then all its projections PuKare centrally symmetric (and centred at 0) so that

minz∈u^⊥vol(PuK)^∗z)=vol(PuK)^∗0)=vol(K^∗0∩u^⊥).

Under this hypothesis, we proved that u→ vol(K^1∗0∩u^⊥) is the restriction to Sⁿ⁻¹ of a norm on ⺢ⁿ. This is Busemann [3] theorem on the sections of convex centrally symmetric bodies, applied toK^∗.

(2) For every convex bodyKin⺢ⁿ, Theorem 1 defines a centrally symmetric convex bodyJ(K) in⺢ⁿby

J(K)= {x∈⺢ⁿ;NK(x)≤1}.

Notice that for everyx₀∈⺢ⁿ, J(K+x₀)=J(K) and that forA:⺢ⁿ→⺢ⁿ a linear isomorphism, we have

J((AK))= |det(A)|(A^∗)⁻¹(J(K)). (3) Ifn=2 and ifRis the rotation by angleπ/2 in⺢², then

vol(P_uK)=h_K(Ru)+h_K(−Ru)=h_K(Ru)+h_−K(Ru),

so that

J(K)= ¹₄R(K−K).

3. The convex intersection bodiesIC(L,z)of a convex bodyL. LetLbe a convex body in⺢ⁿ. For a point z∈int(L), theintersection body I(L,z)of L with respect to zis the centrally symmetric star body in⺢ⁿ whose radial functionρI(L,z) is given for u∈Sⁿ⁻¹by

ρI(L,z)(u)=vol({x∈L;x−z,u =0})=vol(L∩(z+u^⊥)). The bodyC(L) is the star body in⺢ⁿdefined by its radial function

ρC(L)(u)=max

x∈L vol(L∩(x+u^⊥)).

Of course, one hasI(L,z)⊂C(L) for everyz∈int(L). It was proved in [13] that these bodies coincide if and only ifLis centrally symmetric aboutz(the ‘if’ part follows easily from Brunn–Minkowski theorem). We define now theconvex intersection body of L with respect to z∈int(L), which we denote byCI(L,z), by

CI(L,z)=J(L^∗z).

Whenz=g(L) is the centroid ofL, we shall denoteCI(L)=CI(L,g(L)). The radial function ofCI(L,z) is thus given foru∈Sⁿ⁻¹by

ρCI(L,z)(u)=min

x∈u^⊥vol((Pu(L^∗z))^∗x)=vol((Pu(L^∗z))^∗s). In view of Theorem 1, one has

THEOREM5.Let L be a convex body. Then for every z∈int(L), CI(L,z)is a centrally symmetric convex body such that CI(L,z)⊂I(L,z).

REMARKS. (1) It is easy to see that one has for every one-to-one affine map A:⺢ⁿ→⺢ⁿ,I(AL,Az)= |det(A)|A^∗−1

I(L,z)

, as well as CI(AL,Az)= |det(A)|A^∗−1

CI(L) .

(2) In the casen=2, foru∈S¹, denoting byRthe rotation around 0 of angleπ/2, one has

||u||C(L)= ||Ru||K−K

≤ ||u||I(L,z)= 1

||Ru||K−z+ 1

|| −Ru||K−z) ₋1

≤ ||u||CI(L,z)=4(||Ru||K−z+ || −Ru||K−z).

(3) The inclusionCI(L,z)⊂I(L,z) is exact in the sense that their boundaries touch;

there always existsu∈Sⁿ⁻¹such that

vol(L∩(z+u^⊥))=vol((P_u(L^∗z))^∗s).

As a matter of fact, this equality means that the centroid ofL∩(z+u^⊥) inz+u^⊥is atz. To see that suchuexists, defineφ:Sⁿ⁻¹→⺢by

φ(v)=vol({x∈L;x−z;v ≥0}).

Sinceφis continuous, it reaches its maximum at some pointu∈Sⁿ⁻¹. Then, by [15],z is the centroid ofL∩(z+u^⊥).See also [10].

(4) It was proved by Gr ¨unbaum ([10, Section 6.2]) that for every convex body L∈⺢ⁿ, there exists somez0 ∈int(L) such that (n+1) different hyperplanes through z0, with normalsu1, . . . ,un+1, satisfy thatz0is the centroid ofL∩(z+u^⊥_i ). For thisz0, the boundaries ofCI(L,z0) and ofI(L,z0) have at least 2(n+1) contact points.

(5) We have seen above that, in the case whenLis centrally symmetric aboutz, CI(L,z)=I(L,z) and Theorem 5 is nothing else but the classical Busemann’s theorem [1]. Conversely, the following result holds.

PROPOSITION6.One has CI(L,z)=I(L,z)if and only if L is centrally symmetric about z.

It is a consequence of the following lemma.

LEMMA7.Let L be a convex body and z∈L. Suppose that z is the centroid of every hyperplane section of L through itself. Then L is centrally symmetric about z.

Proof.Fix somez0∈int(L),z0=z, and defineF:⺢ⁿ→⺢by F(y)=vol({x∈L−z0;x,y ≥1}). By [15],FisC¹on{F>0} =⺢ⁿ\ {0}and one has fory=0

∇F(y)= ∇F(y),yg({w∈L−z0;w,y =1})

= ∇F(y),y

g({x∈L;x−z₀,y =1})−z₀

. (2)

LetHbe the affine hyperplane in⺢ⁿdefined by

H = {y∈⺢ⁿ;z−z₀,y =1}.

If y∈H, the hyperplane {x∈⺢ⁿ;x−z₀,y =1}passes through z, so that by the hypothesis, one has

g({w∈L;x−z0,y =1})=z, thus, by (2) we get

∇F(y)= ∇F(y),y(z−z0). Now ify,y∈H, one has

y−y,z−z0 =0 and for everyt∈⺢, (1−t)y+ty∈H so that

F(y)−F(y)= ¹

0

y−y,∇F((1−t)y+ty))dt=0.

Thus,Fis equal to some constantconH. Define a functionG:Sⁿ⁻¹ →⺢by G(u)=vol{x∈L;x−z,u ≥0}.

and let

U= {u∈Sⁿ⁻¹;u,z−z0>0}.

Thenu→y(u) :=_u,z^u_−z0 is a one-to-one mapping fromUontoH, and it is easy to check that

G(u)=F y(u)

for everyu∈U.

It follows thatG=conU, and sinceG(u)+G(−u)=vol(L) for everyu∈Sⁿ⁻¹,G= vol(L)−con−U. Now,Sⁿ⁻¹∩(z−z0)^⊥ is contained in the closures of bothUand of−U inSⁿ⁻¹. SinceGis continuous onSⁿ⁻¹,G=c=1−c= vol(L)

2 on Sⁿ⁻¹. The fact thatL is centrally symmetric aboutznow follows by a classical result (see [8]

or [6]).

4. Additional comments and some open problems. We know that althoughC(L) andI(L,z) are not in general convex bodies (by [14],C(L) is convex forn≤3 and by [2], if nis the simplex in⺢ⁿ,C( n) is not convex ifn≥4). HoweverC(L) andI(L,g(L)), whereg(L) is the centroid ofL, arealmost convex, and evenalmost ellipsoids, in the sense that there exist some constantsc>d >0, independent onnandL, such that for everyu∈Sⁿ⁻¹, one has

d

vol(L)³² L−g(L)

x,u²dx ¹₂

≤ 1

maxtvol

L∩(tu+u^⊥) =ρC(L)(u)

≤ 1

vol(L∩u^⊥) =ρI(L,g(L))(u)

≤ c

vol(L)³² L−g(L)

x,u²dx ¹₂

.

In the centrally symmetric case, this was proved by Hensley, and Ball [1] (for sharp constants, see also [19]), and in the general case by Sch ¨utt [22] and Fradelizi [7] (the latter with sharp constants). We have seen thatρI(L,g(L))≤ρCI(L,g(L)). A natural question to ask now is

OPEN PROBLEM1. Does there exist a universal constantC>0, independent on the convex bodyLin⺢ⁿ and onn≥1, such thatρCI(L,g(L))≤CρI(L,g(L))? Of course, in view of the preceding inequalities, an affirmative answer to this question would say that the radial functions ofC(L),CI(L) andI(L,g(L)) are all equivalent (with absolute constants).

Observe that an equivalent way of formulating this problem is the following. LetK be a convex body in⺢ⁿsuch that its Santal ´o point is at 0. Does there exist an absolute constantC>0, independent onnandKsuch that

vol

(P_uK)^∗Puz

≥Cvol

(P_uK)^∗0

for everyz∈int(K) ?

Equivalently, given a convexM⊂u^⊥, with Santal ´o points(M), and a convex bodyK in⺢ⁿ, with Santal ´o points(K), such thatP_uK=M, does

vol(M^∗s(M))≥Cvol(M^∗Pus(K)) for some universal constantC>0 ?

If one could prove that in this situation, for some universal constantc>0, the following is true:

Pus(K)−s(M)∈ c n

M−s(M) ,

then an affirmative answer could be given, using the following lemma.

LEMMA8.Let V be a convex body in⺢ⁿand x,y∈int(V). Then (1− ||x−y||V−y)ⁿvol(V^∗x)≤vol(V^∗y)≤ vol(V^∗x)

1− ||y−x||V−x

n

Proof. One has

vol(V^∗y)=vol(V^∗y+y)=vol

(V−y)^∗

=vol(

(V−x−(y−x))^∗

= (V−x)^∗

1

(1− y−x,z)ⁿ⁺¹dz

because by a formula given in [18], ifLis a convex body with 0 in its interior, one has for everywin the interior ofL,

vol(L^∗w)=

L^∗

1

(1− w,z)ⁿ⁺¹dz. Sincey−x,z ≤ ||z||(V−x)^∗||y−x||V−x, we get

vol(V^∗y)≤

(V−x)^∗

1

(1− ||z||(V−x)^∗||y−x||V−x)ⁿ⁺¹dz. (3) Now, ifLis a convex body with 0 in its interior, and 0<c<1, then

L

1

(1−c||z||L)ⁿ⁺¹dz=nvn Sⁿ⁻¹

||θ||L1

0

rⁿ⁻¹

(1−cr||θ||L)ⁿ⁺¹dr

dθ

=nvn Sⁿ⁻¹

1 n

r 1−cr||θ||L

n_||θ||L¹

0

dθ

=vn Sⁿ⁻¹

1

(1−c)ⁿ||θ||ⁿ_Ldθ= vol(L) (1−c)ⁿ. From which, together with (3), we get

vol(V^∗y)≤ vol(V^∗x) (1− ||y−x||V−x)ⁿ.

Applying the same formula while interchanging the roles ofxandy, one has vol(V^∗x)≤ vol(V^∗y)

(1− ||x−y||V−y)ⁿ.

It is well known (see, e.g. [19]) that there is an affine transformationA:⺢ⁿ→⺢ⁿsuch thatAL=Misisotropic, that is, it satisfies vol(M)=1 and for everyu∈Sⁿ⁻¹,

1

vol(M) _M−g(M)x,u²dx ¹₂

=LM,

whereL_Misthe isotropic constantofM. In that context, Problem 1 is equivalent to OPENPROBLEM2. LetMbe an isotropic convex body. IsCI(M) equivalent to the Euclidean ball (with absolute constant independent onM⊂⺢ⁿand onn) ?

Of course, Problems 1 and 2 are non-trivial only ifL or M are not centrally symmetric. The particular case of the simplex is still open.

OPENPROBLEM3. Let nbe a simplex in⺢ⁿ. Is there a constantcindependent on nsuch that for everyu∈Sⁿ⁻¹

vol( _n∩u^⊥)≤cvol((( _n∩u^⊥)^∗0)^∗s)=cvol((P_u( ^∗_n^g))^∗s)?

Observe that when n is a regular simplex inscribed in the Euclidean ball, since ( n)^∗0= −n n, one has

( _n∩u^⊥)^∗0=P_u(( _n)^∗0)=P_u(−n _n) and thus

vol((( _n∩u^⊥)^∗0)^∗s)= 1

nⁿ⁻¹vol((P_{u n})^∗s).

About Problem 3, one may remark that by affine invariance, we may suppose without loss of generality that n is the regular simplex with vertices e1, . . . ,en+1,|ei| =1, centred at 0=e1+ · · · +en+1. For 1≤i=j≤n+1, one has thenei,ej = −¹_n.

FACT.Let A⊂ {1, . . . ,n+1}be such that1≤k:=card(A)≤n and define uA=

i∈Ae_i

|

i∈Aei| =

n k(n+1−k)

i∈A

ei∈Sⁿ⁻¹. Then0is the centroid of _n∩u^⊥_A.

It suffices to show that 0 is the Santal ´o point ofPuA n. Let

e_l=PuAel, 1≤l≤n+1, E=[e_i,i∈A], andF=[e_j;j∈A].

ThenEandFare linear subspaces ofu^⊥_Asuch that dim(E)=k−1, dim(F)=n−k, x,y =0 for everyx∈Eandy∈F, and

SA:=conv ({e_i,i∈A})⊂EandTA:=conv ({e_j,j∈A})⊂F

are regular simplices with centre of mass at 0. It follows that 0 is the Santal ´o point ofS_AinEand ofT_AinF, when 0 is the Santal ´o point ofP_{uA n}=conv(S_A,T_A). It

follows that 0 is the centroid of _n∩u^⊥_A, which can be described as

n∩u^⊥_A=S_A^∗ ×T_A^∗,

whereS^∗_AandT_A^∗are the polars ofSAand ofTA, respectively, inEandF. A corollary of this result is the following.

PROPOSITION9.For every A⊂ {1, . . . ,n+1}, such that1≤k:=card(A)≤n, one has

||u_A||CI( n,0)= ||u_A||I( n,0).

Moreover, in the particular case when u^⊥∩ n is itself a simplex, one has the following computational proposition.

PROPOSITION10.Let u∈Sⁿ⁻¹, and if u=_n+1

i=1 uiei∈Sⁿ⁻¹ with_n+1

i=1ui=0 and u1, . . . ,un≥0>un+1, then u^⊥∩ nis a simplex and

ρI( n,0)(u)=vol( n∩u^⊥)= 1 (n−1)!

(n+1)ⁿ⁺¹² nⁿ²⁻¹

n 1

i=1(u_i+n j=1u_j) and

ρCI( n,0)(u)=vol(( n∩u^⊥)^∗0)^∗s)= 1 (n−1)!

nⁿ²⁺¹ (n+1)ⁿ⁺₂¹

n1

i=1u_i.

It follows from Proposition 10 thatCI( _n,0) has 2n+2 small faces aroundu= ±e_i, 1≤i≤n+1. Nevertheless, it is easy to check that for such directionsu∈Sⁿ⁻¹ one has

1≤ vol( n∩u^⊥) vol(( n∩u^⊥)^∗0)^∗s) ≤ e

2.

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